In Julia REPL (plz make sure you are running Julia 1.8)
using Pkg
Pkg.build("IJulia")
This command should register the julia-1.8 kernel.
Just to be sure, you can run jupyter kernelspec list to see if julia-1.8 kernel is installed.
Edit: And as @mkitti suggested, please make sure IJulia is using the same python environment as quarto is.