Short answer: The Soviets had already invaded - at German request - by the time this exchange could even have happened
German troop movements in "Soviet Poland" cannot be considered the catalyst for the Soviet invasion. The Germans crossed the original demarcation line discussed in the Molotov-Ribbentrop Pact two weeks before the Soviet invasion. Far from being motivated to secure "their" territory (or even protect the Galician oil fields, which Germany captured on September 14th), the Red Army was determined to wait until the fall of Warsaw. Molotov stalled for more time but gave in to Ribbentrop's urging on September 16th, with the Red Army invading the following day.
The closest historical analog to the described incident took place on September 19th, by which point Soviet forces had already reached Lviv.
Long answer: the quoted opinion is confused about the timeline of events
There are two intertwined questions here that I will attempt to break down and answer:
The answer to both questions is "no".
No regions of Poland were "granted" to the USSR under the Molotov-Ribbentrop Pact, and there was no "Soviet Poland"; instead, the USSR and Germany divided Poland into spheres of influence:
Article II. In the event of a territorial and political rearrangement of the areas belonging to the
Polish state, the spheres of influence of Germany and the U.S.S.R. shall be bounded approximately
by the line of the rivers Narev, Vistula and San.
The question of whether the interests of both parties make desirable the maintenance of an
independent Polish State and how such a state should be bounded can only be definitely
determined in the course of further political developments.
There was no provision for Germany "handing over" any part of Poland to the USSR (because it did not, at the time, hold any part of Poland) and the Soviets knew full well that they would have to invade. The Germans did as well: almost immediately after the German declaration of war, Friedrich Werner von der Schulenburg was ordered by Ribbentrop to inquire about Soviet invasion plans, though Molotov demurred:
советское правительство согласно, что ему в подходящее время «обязательно придется... начать конкретные действия. Но мы считаем, что этот момент пока еще не назрел», а «торопливостью можно испортить дело и облегчить сплочение противников»
the Soviet government agrees that it shall soon have to "take specific action...but we believe that this moment has not come" and "rushing may spoil the matter and help our enemies band together"
Molotov considered it acceptable for troops of either side to cross the proposed Narev-Vistula-San line:
советское правительство допускает, что в ходе операции одна или обе стороны будут вынуждены временно перейти линию интересов, однако это не будет препятствием к осуществлению принятого плана
The Soviet government allows for the possibility that during the operation, one or both sides will see the need to temporarily cross into the other's sphere of influence; however that shall not be an obstacle to implementing the established plan
Germany started its invasion on September 1st, and reached the Vistula by September 3rd. By September 14th, German troops had crossed the Narev-Vistula-San line in multiple places, advancing as far as Brest and Lviv. Comparing this map to the map of oil fields below, it can be seen that the Germans already held the area where the oil was, to the south-west of Lviv.
On September 14th, Molotov informed von der Schulenburg that the Red Army was waiting for the fall of Warsaw to invade:
Учитывая политическую мотивировку советской акции (падение Польши и защита русских «меньшинств»), было бы крайне важно не начинать действовать до того, как падет административный центр Польши — Варшава
Considering the political motivation for Soviet action (the fall of Poland and protecting Russian minorities there) it is critical not to act until Poland's administrative center - Warsaw - falls
Ribbentrop retorted that Warsaw was about to fall, and if the Red Army was not in control of Eastern Poland by that time, "conditions for the formation of a new country" might form. This convinced Molotov to act more swiftly, and the Red Army attacked on September 17th, 1939 (the conclusion of a non-aggression pact with Japan on September 16th likely also played a role).
The incident that the Quora answer refers to appears to have taken place on September 19th, and could refer to one of two possible meetings that happened around the same time, neither of which involved Stalin (between Molotov and von der Schulenburg, or between German military attache Köstring and People's Commissar of Defense Voroshilov). However, the specific assertions in the Quora post are almost all wrong.
The incident kicked off at 4 AM on September 19th, when Soviet tanks entering Lviv clashed with German troops who were already engaged in taking it.
That evening, the Soviet attache in Berlin, Belyakov, reported to Molotov that Walter Warlimont, a member of the OKW operations planning staff, had shown him a map of the German Reich's future borders, which included Lviv, the day prior. When pressed by Molotov, von der Schulenburg responded that the demarcation line on that map was intended to be temporary.
At 3 AM the following day, Köstring met with Voroshilov to resolve the issue of the troops in the field. At this level there was no disagreement about what needed to be done; Köstring arrived at the meeting with existing orders in hand from Hitler to fall back 10km west of Lviv and let the Soviets have it.
While Voroshilov was aware of the Warlimont map, Köstring was surprised to learn about it:
Карта, которую показал начальник оперативного управления Варлимонт Белякову, имела линию границы не в соответствии с договоренностью советской и германской сторон, и она не может считаться линией границы, а только лишь линией, которую должны занять германские войска. На замечание Народного комиссара, что на карте Варлимонта, которую он показал Белякову, была линия границы, проведенная, от Варшавы по Висле и далее к востоку от Львова, Кестринг, явно смутившись и в шутливом тоне сказал, что Варлимонт не политик и, возможно, что он, как работник-нефтяник, соблазнился нефтью, но что из-за этого они не позволят себе нарушать достигнутое соглашение и что это был маленький инцидент.
The map that Warlimont showed Belyakov depicted a line deviating from the one in the Soviet-German agreement. It was not to be considered the border, merely the line that German troops were to advance to. When Voroshilov noted that the depicted boundary ran from Warsaw along the Vistula and then to the east of Lviv, Köstring was surprised, and joked that Warlimont was not a politician and perhaps as an oil-worker he was tempted by the oil there; however this was only a small incident that does not mean Germany will violate the agreement.
Köstring and Voroshilov immediately worked out the positions German troops would fall back to and the schedule of the withdrawal.
Later that day, Hitler sent a proposal for a permanent line to Molotov which would grant a small part of east Galicia's oil fields (up to the town of Turka, indicated on the map below) to Germany; Molotov refused to give up any Ukrainian territory, but ceded Suwałki and its surroundings in compensation.
The Soviet side gave the Germans until September 23rd to leave Lviv and coordinated on a plan of action to avoid further clashes; this plan would then be amended at general von Brauchitsch's request with a further 24 hour delay.
A week later, Ribbentrop did ask Stalin for these oil fields, but was also turned down:
Риббентроп ... выразил надежду, что Советское правительство сделает уступки в районе нефтерождений на юге в верхнем течении реки Сан. Сталин заявил, что «в этом отношении какие-либо встречные шаги со стороны Советского правительства исключены. ... Однако в качестве компенсации Германии были предложены поставки до 500 тыс. тонн нефти в обмен на поставки угля и стальных труб.
Ribbentrop expressed the hope that the Soviet government would make concessions in the area of oil deposits in the south, on the upper San river. However, Stalin ruled out any adjustments in this regard. As compensation, Germany was offered 500,000 tons of oil in exchange for coal and steel pipe shipments.
Therefore we can see that the Quora answer gets almost everything wrong:
The Germans were eager for the Soviets to send in troops, and put political pressure on Molotov to that effect
The Red Army invaded Poland before the described incident took place
The Soviets knew that German troops had been operating in their zone of influence for two weeks before the incident, and this did not motivate them to invade
Without German pressure, the Soviets would have invaded when Warsaw fell anyway - September 28th, technically later but only by 11 days
The map was voluntarily shown to the Soviet attache by an operations officer
The Germans did not attempt to justify the map to the Soviets, and had ordered the withdrawal even before the negotiations around the finalized borders
The Soviets granted the Germans the schedule for the withdrawal that they requested
Stalin was not really involved
It bears noting that the final boundary agreed upon between the USSR and Germany granted Germany considerable additional territory beyond the initial San-Vistula-Narev line (the area between Przemyśl and Lake Solina, as well as all of Lublin Voivodeship, Siedlce Voivodeship, and Podlaskie Voivodeship), so the portrayal of Stalin as a hardline negotiator refusing to yield an inch is also highly inaccurate.