You can do the following:
julia> using CSV, DataFrames
julia> io = IOBuffer()
IOBuffer(data=UInt8[...], readable=true, writable=true, seekable=true, append=false, size=0, maxsize=Inf,ptr=1, mark=-1)
julia> df = DataFrame(rand(1:10, 3, 5), :auto)
3×5 DataFrame
Row │ x1 x2 x3 x4 x5
│ Int64 Int64 Int64 Int64 Int64
─────┼───────────────────────────────────
1 │ 6 10 5 4 4
2 │ 1 9 6 5 3
3 │ 5 4 5 8 4
julia> CSV.write(io, df; quotestrings=true, transform=(col,val)->string(val)) |> take! |> String |> println
"x1","x2","x3","x4","x5"
"6","10","5","4","4"
"1","9","6","5","3"
"5","4","5","8","4"
The trouble is that quotestrings only forces quoting strings (so that when you read back the file numbers are not quoted and correctly parsed) and therefore you need also transform argument to force every value to be written as string.