You have the right idea! There is just an incomplete word there. And, YES, there are different ways of saying this phrase.
First, the correction: "Faz quatro meses que aprendo português." You can see that I added the letter "z" to complete "Faz", and I removed the "o" before "português". The "o" makes the phrase sound like you are "learning the Portuguese" instead of "learning Portuguese".
Second, the other ways to say this phrase are:
-Eu estudo português há quatro meses. (I am studying portuguese for four months now.)
-Estou a estudar/aprender português há quatro meses. (I am studying/learning portuguese for four months.)
Hope this helps!