The alien only knows base 4. The astronaut says base 4 and that he uses base 10.

Think about: What is “4” in base 4?
The alien says, it uses base 1 0, which is 1 * 4 + 0 * 1.

So the term “base 10” (in base 4) is the same like “base 4” in base 10.

The alien does not know, what base 4 is, because you cannot count to 4 in base 4.