Answer: 10 quarters, 9 dimes, 6 nickels, and 5 pennies

How I got the solution:

If you have more than 0 pennies, you need at least 5 of them in order to end up with a multiple of 5 in your total. That means you need at least 6 nickels (you have more nickels than pennies) and at least 7 dimes (you have more dimes than nickels). That means you get a minimum of 1.05 in dimes nickels and pennies. So the maximum number of quarters you can have is 10 otherwise you would end up with more than 3.75.

Let's start with 10 quarters, 7 dimes, 6 nickels and 5 pennies: you get 3.55

You still need to add 20 cents somewhere without breaking the rules.

The only way to add 20 cents without breaking one of the rules is by adding 2 dimes. So you end up with 10 quarters, 9 dimes, 6 nickels, and 5 pennies as your answer.

If you try with 9 quarters, you can't find a solution that doesn't break the rules so the above answer is unique.