Install a keylogger on the pc without him knowing and see what comes up.
that is a criminal offense in most of the civilized world.
the lengths people go to avoid talking to each other, geez.
Not really,especially if it's a shared PC.
And it's not about talking,it's about being sure that you're not ruining a relationship with paranoia and that you have good cause.
After all,the OP didn't report what the texts say.