Part of the story that George and Lenny share and repeat with one another concerns the idea that the two of them are not alone. They have each other to rely on. This is what separates the pair from other iterant (traveling) workers.
Yes, George anticipates future trouble and we can see this anticipation in at least two places. First, even before George and Lenny get to the ranch, George instructs Lenny about where they will meet if something bad happens. George insists that Lenny repeat and rehearse the plan, which emphasizes the notion that George believes trouble is inevitable. He wouldn't bother to make sure Lenny memorized this "just in case" plan, if he didn't really think that there would be trouble.
Also, upon meeting Curley and noticing Curley's attitude toward Lenny, George tells Lenny to watch out for Curly and explicitly says that Curley is going to make trouble and start a fight.