How can you multiply 1230 and 23 with base 4?

You can either turn it back to decimal, calculate it and turn it back to quaternary. Or you can draw a simple times table in Base 4:

x 1 2 3

1 1 2 3

2 2 10 12

3 3 12 13

Then, when you multiply normally, like 1230*20 and 1230*3 you will get 31200 and 10330 respectively. Add then up and you get 102130 in quaternary, or 1180 in decimal.