Up to the DM
If this is going to work is up to the DM. The only content we have about growing Myconids is given in the Monster Manual section about it, which states
Myconid reproduction. Like other fungi, myconids
reproduce by mundane sporing. They carefully control
their spores' release to avoid overpopulation.
So, yeah, having the (reproductive) spores 1 should be enough for them to grow up, as long as they are given conditions to survive (which aren't exactly stated, but mostly nutrients and a regular underdark environment, possibly?)
About how long it will take to grow up, Myconids have 2 stages described, which are the Sprout and the Adult. The Forgotten Realms wikia states the average lifespan of Myconids as roughly 24 years (for all the editions there, from 3e to 5e, which might mean that this is something taken from earlier editions, not described in 5e material). How much of this time is spent as a Sprout and how much is spent as an Adult is up to the DM. How much it takes from Spore to Sprout is also up to the DM.
If you want to base that time in real life fungi, that's out of my expertise, sadly. As far as I know, though, real life fungi live way less than 24 years, so you will need to do some scaling anyway. Either way, for it to become an actual Myconid Adult (which I suppose is the player's intent), it should take some months or even a few years.
1 Myconids have a few other spores, listed in their actions. Pacifying, Rapport, Hallucination and Animating (last two only from Sovereign) spores are not the mundane spores I'm referring to. Again, it's up to you (and the scenario which led to the acquisition of the spores) what kind of spore the rogue got. Note that these mundane spores probably can't be used as poison.