You could just say 질문이 있어요 (I have a question). But that depends on the situation.

So anyways, let's look at yours: 물어 볼게 있어 and 물어 볼거 있어

Both mean "There's something I want to ask you". Literally, they mean "There is a thing (I) will try asking".

The only difference we see is "게" and "거", right? So what we need to understand is the difference between those two things.

게 is simply the contracted form of 것 + 이. 것 means "thing" and 이 is a particle.

거 is simply the colloquial form of 것 ("thing"). So we can see there is no particle included. It is therefore more conversational.

However, to say "there is x", usually (officially) the pattern 이/가 있어(요) is used. So I would say that both are correct, but the former (물어 볼 게 있어) is technically/grammatically correct and the latter is more casual/conversational.

Thanks Jangyook, since you mentioned about the spacing, I would like to know more about the use of spacing in korean, how do I know when to put a space. I tend to just put it where I feel like there should be one based on my shallow experience on reading korean literature. However, I would like to know the accurate way to it. Could you explain how the spacing works in the Korean language? 

Thanks.

Both are we often use. 

But grammtically wrong.  Becasue of the spacing.

물어 볼게 있어  (x) ---> 물어볼_게_있어 (O),  물어볼게_있어 (O)

물어 볼거 있어 (x) ---> 물어볼_거_있어 (O),  물어볼거_있어 (x)

and this spacing grammar is most of Korean don't know about.

I am 100% sure. Many Korean don't know this grammar just because we don't care. 

I also had to use the spell checker. Lmao