It depends on the Hulk. Kinda.
At various times, the Hulk has had:
- A fairly "stable" power level, affected only slightly to moderately by his anger level
- A power level (and even physical body) that increased in direct proportion to anger level
- A power level (and physical body) that decreased in direct proportion to anger level ('Smart Hulk, Savage Banner' – some of the most interesting modern Hulk stories in the modern era)
- Many, many other permutations of the above (The Maestro, various Banner/Hulk splits, Grey Hulk, etc etc).
The Thing, honestly, would have a hard time beating down any of these (Savage Banner excluded)… Nevertheless, The Thing tends to be depicted as being "Stronger than a Hulk at rest", that is, having a more consistent strength that can beat a "barely transformed classic Hulk", and able to stand toe-to-toe against a moderately angry one.
The only problem is... There appears to be no limit to the rage potential of the Hulk, and fighting tends to make him more angry (he just wants to be left alone, after all).
Thus, whether they are fighting each other, or even working together, even if the Thing starts out stronger than the Hulk, that advantage will inevitably give way to the ever-increasing power level of the Hulk, at least until whatever conflict/issue is resolved that allows Big Green to calm down again.