The phrase "a legitimate question to" is correct and usable in written English.
You can use it when introducing a question that is valid or reasonable in a given context.
Example: "Is this a legitimate question to ask during the meeting?"
Alternatives: "a valid question to" or "an appropriate question to".
"That would be a legitimate question to ask," Blower said.
He added: "I do think there is a legitimate question to ask in this modern age.
"I think it was a legitimate question to ask, there's no harm in asking a question," he said.
The prosecutor went on to say that he could not call the co-pilot's actions suicide, but that it was a "legitimate question to ask".
" 'Evan, what are you doing?' is a legitimate question to ask, but the critic never asked me," Mr. Maurer said, recalling a vicious review in a local newspaper.
The scenario was identical: rumours were swirling "on the internet" and Marr felt it had become a legitimate question to ask.
It is a perfectly legitimate question to ask whether Senator Clinton has any experience at all besides being a senator for a short period of time (where she was completely ineffectual in stopping the worst and most unpopular president in US history).
The question of whether non-human animals enjoy it too is a perennial – and scientifically legitimate – question to ask.
When I graduated from college, in 2005, it was a legitimate question as to whether an aspiring journalist should try to get their foot in the door at the best place possible, or go out into the country and work for a smaller paper, learning the ropes by covering courts or cops for a few years.
The McCartney industry is enjoying such a boom that there is a legitimate question as to the necessity of new material.
It's a legitimate question as to why I launched Disruptive Women in Health Care six years ago in the US and, today, will be introducing Disruptive Women UK.
