It seems there may be one interesting encounter which is recorded regarding Marcus Aurelius and the Christian population at the time. Marcus gives some credit to prayer by Christians within his forces to an event during his campaign in the north (here from Apology):
For having cast themselves on the ground, they prayed not only for
me, but also for the whole army as it stood, that they might be
delivered from the present thirst and famine. For during five days we
had got no water, because there was none; for we were in the heart of
Germany, and in the enemy's territory. And simultaneously with their
casting themselves on the ground, and praying to God (a God of whom I
am ignorant), water poured from heaven, upon us most refreshingly
cool, but upon the enemies of Rome a withering(3) hail.
and this apparently caused a declaration before the Senate :
Founding upon this, then, let us pardon such as are Christians, lest
they pray for and obtain such a weapon against ourselves. And I
counsel that no such person be accused on the ground of his being a
Christian. But if any one be found laying to the charge of a Christian
that he is a Christian, I desire that it be made manifest that he who
is accused as a Christian, and acknowledges that he is one, is accused
of nothing else than only this, that he is a Christian; but that he
who arraigns him be burned alive. And I further desire, that he who is
entrusted with the government of the province shall not compel the
Christian, who confesses and certifies such a matter, to retract;
neither shall he commit him. And I desire that these things be
confirmed by a decree of the Senate. And I command this my edict to be
published in the Forum of Trajan, in order that it may be read.
So, it appears that he experienced what he interpreted as a miraculous intervention, enough that he forbade punishment of Christians merely for their faith. Please note the author of the above document is not Marcus Aurelius himself, but the individual mentioned in the question, Justin Martyr, so this writing is obviously 'biased' from a Christian viewpoint.
The view expressed in Apology, does not seem to be held by some later historians however. Gibbon, in his HISTORY OF THE DECLINE AND FALL OF THE ROMAN EMPIRE, discusses this in Chapter XVI: Conduct Towards The Christians, From Nero To Constantine.—Part V:
The edict of Marcus Antoninus is supposed to have been the effect of
his devotion and gratitude for the miraculous deliverance which he had
obtained in the Marcomannic war. The distress of the legions, the
seasonable tempest of rain and hail, of thunder and of lightning, and
the dismay and defeat of the barbarians, have been celebrated by the
eloquence of several Pagan writers. If there were any Christians in
that army, it was natural that they should ascribe some merit to the
fervent prayers, which, in the moment of danger, they had offered up
for their own and the public safety. But we are still assured by
monuments of brass and marble, by the Imperial medals, and by the
Antonine column, that neither the prince nor the people entertained
any sense of this signal obligation, since they unanimously attribute
their deliverance to the providence of Jupiter, and to the
interposition of Mercury.During the whole course of his reign,
Marcus despised the Christians as a philosopher, and punished them as
a sovereign.
