Can I conditionally add a where() clause to my knex query?

Yes. Use modify.

As applied to your example:

router.get("https://stackoverflow.com/questions", function (req, res) {
    knex('questions')
        .select('question', 'correct', 'incorrect')
        .limit(50)
        .modify(function(queryBuilder) {
            if (req.query.param) {
                queryBuilder.where('somecolumn', req.query.param);
            }
        })   
        .then(function (results) {
            res.send(results);
        });
});