Is it wrong to catch a wild pigeon to keep it as pet?

Of course it is.  If you were a wild bird and suddenly, someone grabbed you and imprisoned you in a cage, for the rest of your life, how would you feel?

It is cruel to keep any wild animal as a pet.

Attempting to keep a wild bird as a pet is a horrible idea in the vast majority of cases, and depending on where you live, it could also be illegal. In the United States, it is against the law to keep any sort of wild native bird captive, and anyone who is caught doing so could be charged with a felony.

Keeping Wild Birds as Pets

www.thesprucepets.com/keeping-wild-bird-as-pet-consid…

Yes.  It would be very wrong to catch a wild animal and imprison it for the rest of his/her life just for your own enjoyment.  Remember the movie Misery?  It's basically like that.  It's wrong to do that to any bird even if domesticated.