"quite accept" is grammatically correct and can be used in written English.
It means to fully or completely accept something. Example: Despite their initial hesitation, the committee quite accepted the proposed changes to the company's policies.
Exact(31)
I can't quite accept this.
I didn't quite accept that, but I thought 'Why not?
Susan Carithers, a member since birth, could never quite accept integration as God's will.
When the show ends, you can't quite accept that the conversation is over.
Segments of the religious right might finally make peace with, if not quite accept, Mormonism's various heterodoxies.
In the end, under some protest, she manages to agree to life in Gopher Prairie, but not quite accept it.
Similar(27)
The rest is not quite accepted history.
The south never quite accepted the legitimacy of national rule.
I have never quite accepted that she is.
I was already virtualised without quite accepting it and receiving the full benefits.
"And I'm not quite accepted in the 'literary community' because I write mystery novels".
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com