Bash CLI remove quotes from output of a command

You can use the tr command to remove the quotes:

my_json=$(cat ~/Downloads/json.txt | jq '.name' | tr -d \")

In the particular case of jq, you can specify that the output should be in raw format:

   --raw-output / -r:

   With this option, if the filter´s result is a string then  it  will
   be  written directly to standard output rather than being formatted
   as a JSON string with quotes. This can be useful for making jq fil‐
   ters talk to non-JSON-based systems.

To illustrate using the sample json.txt file from your link:

$ jq '.name' json.txt
"Google"

whereas

$ jq -r '.name' json.txt
Google