While what constitutes a "best" spot is subjective, here are some popular options: Icho Namiki
This road is located in Tokyo, between the Gaienmae and Aoyama-Itchome stations.
The best time to visit is from mid-november to early december.
Rikugien garden
While this doesnt feature only Ginkgo trees, you can enjoy the view in this traditional japanese garden.
Nishi Honganji
The temple grounds feature many Ginkgo trees, plus you get so visit one of the biggest temples around.
The lesson here is that there aren't many dedicated spots for Ginkgo trees, but they are more or less omnipresent in Japan.
Visit during the end of november
and you are sure to get your fill of beautiful colors.
Here is a resource page for other spots in Tokyo.
And here is the same page for Kyoto.
travel.stackexchange.com/users/43756/JS Lavertu
Visit Toyko's Shinjuku Gyoen National Garden to experience the ginko tunnel in the Meiji-Jingu Gaien, the golden canopy and carpet the most glorious in the autumn.
travel.stackexchange.com/users/30343/Giorgio
You could visit the biggest Ginkgo biloba tree in Japan.
It is the Kitakanegasawa no Ichou .
It has a girth of about 22m, and is estimated to be over 1000 years old.
In terms of girth, it is probably the second biggest tree in Japan (after the Kamoh no Ohkusu Camphor tree).
Photo By Aomorikuma, licensed as CC-BY-SA-3.0, via Wikimedia Commons
It is near Kitakanegasawa Station , in Aomori Prefecture, at the northern end of Honshu.
travel.stackexchange.com/users/7811/vclaw
There are Ginkgo trees in a lot of areas of Japan.
So it depends a lot on where you want to go otherwise.
Since the other answers focus on Tokyo
I will just throw some answers for the Kansai area.
Osaka:
There are many Ginkgo trees lining Midosuji which is the central street going from Namba to Umeda which is a nice stroll.
There are also Ginkgo trees as well as many other tree's such as maple in the Osaka castle garden which makes for a nice view.
Kyoto:There are many places in the central town and around temples.
Such as kiyomizu-dera.
My favorite place in Kyoto for Autumn colors is Arashiyama.
You can also take the Torokko which is a train that goes through the valley where you can see a lot of foliage.
Then you can also take a boat ride on the river to get back to Arashiyama Nara: Nara park contains a lot of Ginkgo trees as well.
travel.stackexchange.com/users/35892/Johan Gronberg
There is also a famous gingko tree in Tokyo in the University of Tokyo (accessible to public).
travel.stackexchange.com/users/49854/Guillaume
I was wondering where is the best place in Japan to see the Ginkgo Biloba trees. My friend says there are many in and around Tokyo.   Are there many trees of this kind in other areas (e.g. Kyoto etc)? Or would they appear only in specific places...   It would be a shame to go to Japan and not see these beautiful trees!!
Are Ginkgo trees in Japan rare and if so, where can I find them?
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<japan><sightseeing><nature-and-wildlife>
travel.stackexchange.com/questions/76758
travel.stackexchange.com/users/41168/John Smith 
Visit Toyko's  Shinjuku Gyoen National Garden  to experience the ginko tunnel in the Meiji-Jingu Gaien, or the famous gingko tree in Tokyo in the University of Tokyo. Osaka Castle garden also has gingko and other trees, while Kyoto has gingko trees in many places, especially Arashiyama and Nara Park. Gingko trees are found all over Japan. In Tokyo they are found in the Shinjuku Gyoen National Garden and Tokyo University. Otside Tokyo Osaka and Kyoto are good places to see Gingko trees.
Gingko trees in Tokyo can be seen in Shinjuku Gyoen National Garden and the University of Tokyo. The biggest Gingko tree in Japan can be visited There are Gingko trees in other areas of Japa. These include Osaka castle garden, Arashiyama in Kyoto and Narra park.