I would argue Yes
I have collected several pieces of evidence which, collectively, I think could be construed as an indication that Q considered Picard to be his 'friend' (although the feeling probably wasn't mutual).
In early 2367, Q defied orders of the Continuum and interfered with
the Enterprise-Ds encounter with Locutus of Borg by keeping the
make-shift deflector dish weapon developed by Geordi La Forge from
overloading and destroying the Enterprise. This consequently allowed
the Enterprise to later rescue Picard and save Earth from
assimilation. Q felt responsible for Picard's assimilation and was
willing to face any consequences the Continuum saw fit to set upon
him. (TNG short story: "Civil Disobedience"; TNG episode: "The Best of
Here we have an instance (albeit non-canon) of Q putting himself in potential danger with the Q Continuum because he felt responsible for Picard's being assimilated - I would see this as an act of loyalty and friendship.
Soon after, when Picard had been critically injured, Q gave him look
at his past and a chance to correct what Picard considered to be
mistakes in his life. In the end, Picard realized that all of his
decisions, even his mistakes, were what turned him into the man he
was. (TNG episode: "Tapestry")
One of the things about friends is that they are there to point out our mistakes but not to make fun of us. Although Picard was sure that he would have been happier had he not made the mistake of taking on the Nausicaans, Q discreetly pointed out to him that making that mistake turned out to be one of the best decisions of his life. Now, by 'discreetly' I know that you're probably thinking that sending one back in time isn't a discreet method, but when you think about how Q acted during that episode, he wasn't directly telling Picard about the importance of that 'mistake', but let him discover the truth in a conducive environment.
In 2366 the Continuum, believing that Q had stepped over the line in
introducing humanity to the Borg, de-powered Q and at Q's request,
sent him to live out his days as a human on the Enterprise. Q had
actually requested to be sent to the ship, as its crew were, for
better or worse, the only creatures in the cosmos that came remotely
close to being his "friends" (TNG episode: "Deja Q")
This is one of the best pieces of evidence that Q considered the crew as a whole to be something close to his friends. Consider, though, that Q's main interactions with the crew were with Picard up to this point, so we could interpret this as Q's 'admission', if you will, that Picard was something like a friend to him.
Later in 2367, Q returned to the Enterprise to "properly" thank
Captain Picard for his role in helping him regain his standing in the
Continuum. (TNG: "Qpid")
This also suggest to me that Q felt there was a positive relationship between Picard and for Q to actually be showing that he was grateful for something like this indicates to me he saw Picard as his 'friend'.
So, in summary, yes, based on the evidence I've presented above, I would think that Q considered Picard to be his 'friend', or something close to it.