Sorry to revive an old thread but I finally did this quest for the first time after refusing it several times and because some messages here helped me, I think that a bit of contribution might be helpful also to others.
About duration:
In Qi's Walnut Room, the quest was indicated with a 28 days duration, which is the maximum timeframe for this quest. However, once taken, the quest diary displayed only 14 remaining days because we were already in the middle of winter. The duration indicated in Qi's Walnut Room seems misleading, which can explain some comments on this thread about a shortened duration.
Nonetheless, I accepted the challenge though I never did this quest before.
I was able to complete it in 14 days.
Strategy
Keys to success are: time management and propagation because growing 500 fruits of something that you have no reliable source for the seeds is a real challenge.
Day 1
On the first day, I went to the sewers to change farming profession and took "agriculturist". I also bought many Deluxe Fertilizers at Pierre's (faster that crafting them). I went straight to 400 of them because I was aiming at 500 fruits and I was too lazy to do the maths to get the minimum required amount.
With both, time to grow Qi Fruits was cut down from 4 days to 2 days, which is really important because we need to manage time as much as possible.
Still on the first day, I prepared a crop of about 40 spots, watered and with Deluxe Fertilizer and sprinklers for the next days. I also crafted Seed Makers to have at least 5 of them (and I think more is better) in order to reduce waiting time when converting Qi Fruits into Qi Beans.
All the remaining time of the day was spent in Skull Caverns bombing rocks (and killing nearby monsters) in order to find Qi Beans. Why Skull Caverns? Because time slows down in Skull Caverns (as noted above), so this allows to do more actions in a given amount of time and thus to find more beans per time slot. Also, there are often clusters of rocks, that helps to get more possible items (and thus Qi Beans) for a single action: bombing. The run was made with speed buffs: Triple Shot Expressos and Spicy Eels. High magnetism helps to gather items without moving too much in levels.
I aimed to get 20 to 40 Qi Beans for this run (I got the lower end, unfortunately). I returned to the farm no later than midnight to have time to plant the beans in my previously prepared crop.
Day 2
On day 2, I did only the most urgent chores in the farm and spent the day in Skull Caverns for Qi Beans and returned to the farm early enough to plant the beans before 2:00am.
Given the short time window (14 days), it is very important to plant Qi Beans the day they are found.
Day 3
On day 3, the first crop of Qi Fruits was ready and all fruits went to Seed Makers. Basically, this means that every two days, one Qi Beans produces two Qi Beans (on average). I prepared a crop big enough to hold all the new Qi Beans (watered, fertilized and with sprinklers), did minimal chores on the farm and went to Skull Caverns again for more Qi Beans.
Following days
I followed this routine for more days and stopped extending the crop size once it reached about 350 spots. At this point, once I have created enough new Qi Beans through Seed Makers to fill all the spots freed in the crop after the daily harvest, I hoarded all remaining Qi Fruits.
Last days: last friday to sunday
Obviously, no more Qi Beans are planted after the last friday on the quest, which means that all Qi Fruits harvested and all Qi Beans found during the last saturday and sunday must be kept.
I did not sell a single Qi Fruit before the last sunday when I sold 500 of them. The remaining ones (with a few additionnal Qi Beans found on the last two days) were sold to Pierre in order to be bought back the next days (not done yet).
Conclusions
So, with good preparation and dedication, this quest can be done in 14 days (even less as I did not start on the 1st monday) while laying ground to do it easier on next occurrences.
Surprisingly enough, I did not have too much things to clean up/handle/repare in the farm after two weeks on minimal chores. The fact that it was winter helped, though. Furthermore, the quest was not a money sink (quite the opposite) due to the heavy bombings in Skull Caverns that earned me many, many ores and gems.
I know that people also use fishing for finding Qi Beans through maximising treasure chests finding due to high ratio of Qi Beans in chests. I tried this strategy but I got far less Qi Beans than with a Skull Cavern run. I believe I was unlucky with the treasure chests though.
