Muslim trader presence certainly existed in the Byzantine Empire.
I'd split my answer into three parts:
Quoting from the main source:
Muslim presence in the Eastern Roman Empire can be traced back to very foundation of Islam in the 7th century but at some later point the Muslims were regarded as a quasi community and appropriate institutions emerged, probably in the context of some treaties that had temporarily suspended the constant confrontation between the armies of the Byzantine Empire and the Caliphate.
Among the provisions for the comfort of the Muslims in the Byzantine capital, the foundation and operation of a mosque for the practicing of the Muslims seems to be most characteristic.
The earliest information on the existence of a mosque dates to the early 10th century and can be collected from letters of patriarch Nikolas Mystikos to the Abbasid caliph Al-Muqtadir of Baghdad, which date more precisely in the period of the patriarch's regency, after the death of the emperor Alexandros (913) until the enthronement of Romanos I Lekapenos (920). (Jenkins, R.J.H.- Westerink, L.G. (eds.), Nicholas I. Letters (Washington DC 1973), no. 102.)
(The mosque was probably built for Arab prisoners of war captured by Byzantines)
This correspondence, where the mention to the mosque of Constantinople is to be found, took place on the occasion of the closure of Christian churches within the Abbasid territory, which we are informed that was due to rumors circulating in the Abbasid capital, concerning the alleged closure of the mosque in Constantinople. The patriarch denied these rumors, and at the same time he protested for the closure of the Christian churches. The date of the composition of these letters constitutes a terminus ante quem for the foundation of the mosque of Constantinople, which however cannot be determined more precisely.
Another letter by Nikolas Mystikos to Caliph of Baghdad is described in Muslim perception of other religions by Professor Jacques Waardenburg which shows attempts in forging peace between the two constantly at-war neighbors:
Two Sovereignties, That of Arabs and of Byzantines, surpass all
sovereignties in the world, like the two shining lights in the
firmament. For this one reason, if no other, they should be partners
and brethren. We ought not, because we are separated in the ways of
our lives, our customs and our worship, to be altogether divided nor
should we deprive ourselves from communication with one another in
default of meeting each other in person. That is the way we ought to
think and act, even if no necessity of our affairs compelled us to it.
This letter shows the seriousness in the efforts made to reconcile the two rivals and one can speculate that such efforts may have resulted in treaties bringing more rights to their brothers in faith in their rivals' territory.
The presence of Muslim merchants in Constantinople was continuous at the city's heyday, from the 9th century (if not earlier) up to the 12th century. Whether the presence of Muslim merchants was accompanied by a permanent installation of some of them (as it happened with the Venetians and the Genoese) is uncertain, but even their transitory presence was continuous and regularly renewed. Moreover, apart from the Arab merchants, by the 11th century the Muslim presence of Constantinople also comprised the Turkish element. A poem in the vernacular by John Tzetzes (mid-12th century) is indicative regarding this element, since it mentions that a great number of languages spoken in the markets of Constantinople, including the Arabic and the Turkish. (Ζακυθηνός, Δ.Α., Βυζαντινή Ιστορία, 324-1071 (Athens 1972), pp. 386, 488.)
The Poem is transcribed below,as it was (Which is anti-Semitic but it serves our purposes as it mentions Arabic, Persian (At that point Muslim nation) and Turkish), From the book Byzantium: The Surprising Life of A Medieval Empire:
One finds me Scythian among Scythians, Latin among Latins,
And among any other tribe a member of that folk.
When I embrace a Scythian I accost him in such a way:
"Good day, my lady, good day, my lord:
Salamalek alti, salamalek altugep."
And also to Persians I speak in Persian:
"Good day, my brother, how are you? Where are you from, my friend?
Asan khais kuruparza khaneazar kharandasi?"
To a Latin I speak in the Latin language:
"Welcome, my lord, welcome, my brother:
Bene venesti, domine, bene venesti, frater.
Where are you from, from which theme [province] do you come?
Unde es et de quale provincia venesti?
How have you come, brother, to this city?
Quomodo, frater, venesti in istan civitatem?
On foot, on horse, by sea? Do you wish to stay?
Pezos, caballarius, per mare? Vis morare?"
To Alans I say in their tongue:
"Good day, my lord, my archontissa, where are you from?
Tapankhas mesfili khsina korthi kanda," and so on.
If an Alan lady has a priest as a lover, she will hear such words:
"Aren't you ashamed, my lady, to have a love affair with the priest?
To farnetz kintzi mesfili kaitz fua saunge."
Arabs, since they are Arabs, I address in Arabic:
"Where do you dwell, where are you from, my lady? My lord, good day to you.
Alentamor menende siti mule sepakha."
And also I welcome the Rus according to their habits:
"Be healthy, brother, sister, good day to you.
Sdra, brate, sestritza," and I say "dobra deni."
To Jews I say in a proper manner in Hebrew:
[Anti Semitic bile that I have omitted due to its irrelevance and because its nonsense]
So I talk with all of them in a proper and befitting way;
I know the skill of the best management."
For complete linguistic and historical analysis of the said poem, see this excellent post.
What must be noted here is that until then, the Muslim presence was mainly Arab. Turks did not appear on the steering until end of the Latin Empire and beginning of Turk dominance on traditionally Muslim domains of Levant and Near East. Turkish Sultanates such as Seljuqs of Anatolia did exist before that but at that time they were mere settlements and not the real driving power in Islamic world.
By the time the House of Osman (Ottoman dynasty) had established itself firmly in Anatolian Beyleks, Turkish Muslims in Constantinople had formed a regular Byzantine community. It is implied by the demand of Sultan Bayazid to appoint a Turkish judge in the city, who would attend to the judicial affairs of Turkish resident & subsequently in the description of the defensive measures taken by the Byzantines before the siege of 1453, when they sealed the gates of the walls and arrested any Turks who were caught inside the city. (Bekker, Ι. (ed.), Michaelis Ducae Nepotis Historia Byzantina (Bonn 1834), ΧΙΙΙ p. 49, XV p. 56 and XXXIV pp. 244-245.)
My Own Notes
The Fatmids of Egypt also confirmed presence of such a mosque in their official correspondence between Egyptian officials
but I can't seem to find a reference for that. I found the reference. It is mentioned in "The Fatimids and Byzantium, 10th–12th Centuries" by Yaacov Lev where he says it happened in reign of Basil II.
It happened after Fatmid Caliphs of Egypt succeeded in their diplomatic mission to Byzantium in convincing the Emperor to supplant the sermon given in name of the Abbasid Caliph of Baghdad in Mosque of Constantinople and issue it in name of Fatmid Caliph of Cairo instead. (This was important since in Islamic world, it was a sign of legitimacy. The ruler in whose name the Sermons were issued was considered the Overlord of his Domains and whole Islamic world. The Byzantines accepting Fatmids for that role was great diplomatic triumph for them. For details of Fatmid-Abbasid rivalry, see Shia-Sunni Schism of Islam).
Contrary to what our esteemed member SJuan states, Arab Mosque is not the mosque built by Byzantines. It was built as a Church and later converted to mosque by the Ottomans while the Mosque in question was built as a Mosque and operated as a Mosque by Byzantines, not Ottomans. But we sadly do not know which mosque it was other than that It had existed since the Times of Abbasids and was restored after liberation of Constantinople from Latins.
We don't know if it exists today but I think it doesn't, purely on the grounds that if such a significant monument had survived till our times, Turkey would have made it a tourist destination as they did with other sites of such significance and we would have had more information about it other than some antique scrolls.
Another misbelief is that safety and function of mosque was tied to some treaty forged in war. While it is true that some treaties might have been involved, it was however the norm of Muslim Empire (Caliphate) in the East and Byzantine Empire in the West to tie the freedom of each others religion to condition of their own faith in their rivals' country. As explained above, Abbasid Caliph closed the Churches when he got the apparently wrong intel that Mosque in Greece had been closed and therefore proceeded to reciprocate the act. Byzantines would also employ such tactics in negotiations.
We have established that at least one such instance of such a mosque existed. There might have been more but I doubt it since Muslim traders normally ventured only to thriving markets of Constantinople for trade so it does not make any sense for Byzantines to build mosques for a foreign minority in Greek mainland/hinterland when there was going to be very rare occurrence of visitors coming. Not to mention inhabitants of remote areas would be less tolerant of such acts than compared to populace of a metropolitan city like Constantinople who were accustomed to seeing foreign people, their culture and their rites.
In conclusion however, The treatment of Muslims in Byzantine Empire or Treatment of Christians in Arabian/Turkish Empires varied largely depending on time and relations between the two states. E.g. Muslims enjoyed more rights in Byzantine empire when Beyezid I ruled Ottoman Empire than in 1453 during reign of more belligerent Mehmet II (Subsequently given the epithet 'The conquerer'). To discuss the rights of Muslims in Byzantine Empire for a period of almost 7 centuries from 600s - 1453 would be too broad. It would help if you were to edit your question and restrict it to a certain time period.