My 5 yo grandson is a lot like this. Not thirsty AND prone to constipation. Lots of fiber is great, but can actually be constipating if not accompanied by water.
When I spend a day with him, I do what his parents do, and hold a straw up to his mouth several times during an hour of play, encouraging him to sip often. We do this a couple of times a day. He's more likely to drink if he doesn't have to stop his game, and will often drink more while he's distracted. He can get several additional ounces that way. Plus, at some point during the day, we have a drinking race, to see who can finish a glass of water first. He enjoys that, and I get hydrated, too.