Although the genre of Quiet Storm has been around since the mid-1970s, I only became aware of it when Anita Baker perked up my ears with "Sweet Love" in 1986. I had assumed that it was a sub-genre within the larger one of R&B, but according to Wikipedia, it's considered to be a "super-genre" enveloping "... contemporary R&B, jazz fusion and pop music that is characterized by understated, mellow dynamics, slow tempos, and relaxed rhythms".
If that is indeed the case, then I would consider City Pop in Japan to be along the same lines since it brings together genres such as soul, disco, jazz, sophisticated pop, etc. Getting back to America, though, Quiet Storm has a lot of singers included such as Sade, Luther Vandross and Smokey Robinson, but for me, I think my ambassador will always remain the divine Ms. Baker.
Various aspects of R&B have popped up within Japanese music over the decades. Of course, there was City Pop in the 70s and 80s, and then the parallel rise of hip-hop and soul in the late 90s going into the 00s. However, I don't think Quiet Storm ever got its representation through a large number of singers and/or groups. Yasuhiro Abe（安部恭弘）is perhaps one crooner who can be considered to be a Quiet Storm singer. As for a band, I often think of Sing Like Talking.
The other one who comes closest in reminding me of Anita Baker is singer-songwriter Toko Furuuchi（古内東子）who is celebrating her 25th anniversary since she released her debut single in 1993. Whenever I want to relax to the sounds of light soul and pop, Furuuchi would be the one to head for. And what better song title than "Daijoubu" (It's Alright) to provide solace?
This was her 10th single which came out in July 1997. I was actually viewing her collection of videos last night on the computer when "Daijoubu" popped up; I had completely forgotten about this one and all of the nostalgia juices started flowing again. Those strings and horns put me quite at ease. Furuuchi may not really get out of a certain range of music (can't really see her tackling rock or straight-ahead pop), but when one of her songs crops up once in a while through a compilation or on YouTube, it's enough for me to sit back and enjoy.
"Daijoubu" was also a track on her 6th album "Koi"（恋...Love）from August 1997. It peaked at No. 2 on the Oricon weeklies. Furuuchi hasn't really released anything since the earlier years of this decade, and it's understandable since she is now enjoying motherhood with her son. But according to her J-Wiki article, she did release a live album and cover album in 2016.
Not sure if it will happen but it would be really nice if an equivalent of a Quiet Storm arrived on Japanese shores. Perhaps one really subtle version has already landed through groups such as microstar, Blue Peppers and Blu-Swing in the past few years.