The only way I can think of to say "Thank you" without using the word "thank" is "I'm grateful to you." For a slightly different meaning, you could say "I owe you one." This means that you "owe" the other person something in repayment, but usually not literally "owe".
Instead of saying "You're welcome," you can also say "My pleasure." This has a slightly different meaning, but is commonly used instead of "You're welcome." "My pleasure" is similar to saying "It gave me pleasure to do it." (Whatever "it" is.)