When Sirius first set out to check on Peter, it wasn’t suspicion, it was a pre-existing arrangement. We’re never given details of the arrangement, but since Voldemort would have to go through Peter to get to the Potters, it seems natural that Sirius would want to check he was safe.
This is the relevant passage:
“I persuaded Lily and James to change to Peter at the last moment, persuaded them to use him as Secret-Keeper instead of me… I’m to blame, I know it… The night they died, I'd arranged to check on Peter, make sure he was still safe, but when I arrived at his hiding place, he’d gone. Yet there was no sign of a struggle. It didn’t feel right. I was scared. I set out for your parents’ house straight away. And when I saw their house, destroyed, and their bodies… I realized what Peter must’ve done… what I'd done…”
— Prisoner of Azkaban, chapter 19, The Servant of Lord Voldemort
He only became suspicious once he reached Peter’s hiding place, and realised he wasn’t there.