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I found two chess puzzles on chessvariants.com, Queens's Quadrille and Hippodrome.

I'm thinking of creating random Queen's Quadrille and Hippodrome puzzles with a computer program.

My question is can I always be sure that a randomly generated Queen's Quadrille or Hippodrome puzzle is solvable?

Should I additionally check if it is solvable after generating a random puzzle?

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Are you asking whether all Queen's Quadrille and Hippodrome problems are solvable? They are not. There is a discussion on the Hippodrome page which you link to. Here is an example of a starting position which is valid both for Queen's Quadrille and for Hippodrome, and where no moves are available:

        5bbQ/4BRrB/4krRK/4NNnn/8/8/8/8 w - - 0 1

Perhaps you could consider a version of Hippodrome where the player gets to choose which of the 12 non-knight pieces is removed. Then it seems more plausible that all positions are solvable.

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