That’s a great question. Jesus (who is God the Son, one of the three persons of the Trinity), certainly did suffer in his humanity. When we pray the rosary, we remember some of these sufferings in particular:
- The agony in the garden of gethsemane, as Jesus anticipated the horrors that were about to come upon him, while his friends were unable to even stay awake to offer him emotional support in his grief;
- The scourging at the pillar, where Jesus was violently whipped.
- The crowning with thorns, where soldiers (created beings who Jesus loved with all his heart) mocked him, and spat on him, and scorned him. Covering his eyes and striking him, telling him to “prophecy” and say who struck him.
- The carrying of the cross. So many long, heavy steps. Falling, and probably wanting to stay down, and just be killed with a spear on the spot. But getting back up, and continuing to walk all the way to where his death was going to be conducted by an instrument of torture.
- The crucifixion. This innocent man, this perfectly good man, who loved every single person who was doing this to him, was laid down on this heavy cross, and had heavy nails hammered through some of the most sensitive parts of his body: his hands (where our nerves are clustered for sensitivity in touching things), and his feet. Then the cross was lifted up, and his own body weight dragged down against those nails. Breathing may also have been a challenge; it often was, depending how they hung them.
I could go on. But yes, God suffered for us, in the person of Jesus Christ, the Son.
If you’re asking whether he still suffers, I think that’s a question I’d defer to the experts (e.g. Catholic theologians and professional apologists). I’m pretty sure the general answer is that since the resurrection, no, God does not suffer. His suffering is over (temporally, although in a sense all times are equally ‘now’ to God, so… that’s a possibly interesting question, about whether in a sense His own suffering will always be before Him), and He is now in glory at the right hand of the Father.
At the same time, I’m pretty sure some saints (e.g. St. Faustina) have reported private revelations where they seem to suggest Jesus uses language with them of feeling grieved over human behaviour, and that sounds like a kind of suffering. But private revelations are NOT binding on the faithful to believe, so I wouldn’t rank that too highly.
If you’re hungry for learning about God (which is fantastic!) I’d highly recommend checking out Catholic Answers youtube videos (they have a (videotaped, so you can see them while they talk) radio program where professional theologians and apologists answer caller questions live on air, and you can search questions by topic, to find answers on a topic you’re interested in)
