No, it is always na zahradu. Rarely, in specific circumstances, there is "do zahrady".
But zahrada is not a general park, it is someone's garden, often in the backyard.
Some parks in Prague do have zahrada in their name but they used to be private aristocratic gardens as well.
We simply do not say "jdeme do zahrady" on its own. It is not a preposition that is used in this situation. There is no point searching for any logic in it. English prepositions are often not logical either.
You can use "do" for institutions or parks "do Kinského zahrady", "do zámecké zahrady", but not for your garden where you grow flowers or vegetables.