- Here probably arises a pertinent question as to why he would want to be a bio-engineer now, if he didn't like mathematics at school?
In English, the typical order is subject-verb, and not verb-subject. Switching them: "arises a pertinent question" sounds unusual.
"Here" and "now" seem redundant. They are both saying the same thing. Remove one of them.
"Probably" here makes the writing casual. Sometimes that's ok. If it's intended to be more official, such as an academic paper, or an article, you should perhaps take a firmer stance and not qualify it.
- Here probably the pertinent question arises as to why he would want to be a bio-engineer now, if he didn't like mathematics at school?
Similar feedback.
- Here probably the/a pertinent question arises, "Why would he want to be a bio-engineer now, if he didn't like mathematics at school?
Similar feedback. This is probably the best of the three sentences.
You can't really analyze a sentence like this in isolation. For just simple grammar, okay. But for overall word-choice and balance of the phrasing, the context is important. You should post the preceding paragraph, and subsequent paragraph. Or have a link to an entire page of text.
Suggestion:
- Here a pertinent question arises, "Why would he want to be a bio-engineer, if he didn't like mathematics at school?"