When I got this quest, I was in Era 1, with all three of Era 1's three science buildings researched (Public-private partnerships, magnetic field generators, and the exoscience stations specialization).  I was concentrating on exploring and building at the time, so I didn't have any military ships other than scouts, and those were well away from my core systems at the time.  So I wasn't sure if I could build 4 ships and get them to the two systems it wanted them in within 10 turns - and I'd rather have the reward for the science buildings option, so I chose that option.

But, I did have enough production to finish all three of those buildings in 6 or 7 turns on my home system (I think one was already done), and in 9 turns on another system.  It took me several reloads from autosaves to successfully complete this quest, and I finally ended up having to delete several items from my production queue, one of them almost complete, in order to succeed, and then ignore the empty build queue errors that it gave me for those few turns.

Also, the the mouse-over text that provides more detail about the quest completion steps on the quest says that you will fail it if you complete any ship or non-scientific improvement, but I tested that by putting a non-science building as the next thing in my queue, but that wouldn't be complete until after the 10 turns were up, and that made the quest fail.  So either that will need to be re-phrased somewhat, or change the quest's real conditions so they match the text - so you can put some non-science things in the queue, and make progress on them, but you only fail if those items are actually finished.

So, maybe there's some way to do some testing before choosing to grant this quest to see if the player has enough science infrastructure left to complete?

I would recommend adding Cerebral Reality to the list of allowed buildings for this quest - that one's bonus is half science, and flavor-wise, managing/using that building is a thing that would be attractive to young Sophons.

Perhaps this should allow exploration-hull ships to count as science-focused improvements?  I was down several scouts to pirates at that point, so building more would not have been wasted build queue time at that point in the game.

Perhaps this quest can temporarily unlock access to Planet Cracker Lab/Factory while the quest is going, so that even if there's no science improvements to complete, you can have a planet doing something somewhat productive while it waits for the quest to complete?

The militarist option for that quest is just as gimmicky. The only way I managed to complete it the first time, was to design a new ship with no modules (except for 1 non-symmetric gun just in case) costing only 90 production. Took 6 turns total but it would have been unrealistic to finish fair and square building regular ships at that stage of the game. Even then, the reward was only for a +20 approval building with no upkeep (good!) that costs 1000 production (bad! can only be built much later) and can only be built once in the empire (very bad!). 

As it currently stands, that quest does seem a bit of a trap that should be avoided in the early game. It derails development too much.