how to convert from hexadecimal to base 4 without going through base 10?

Answer 1

Convert each digit individually.

Example:
D18

8 = 2 * 4^1 + 0 * 4^1 = 20 in base 4
1 = 0 * 4^1 + 1 * 4^0 = 01 in base 4
D = 3 * 4^1 + 1 * 4^0 = 31 in base 4

So D18 in base 16 = 310120 in base 4

Answer 2

To convert directly from to base 4 is easy because 4² = 16.

Just convert one digit at a time from base 16 into two digits in base 4 (use leading zeros if necessary). Here is an example.

(Hexadecimal) 6A3 = (Base Four) 12 | 22 | 03 = 122203