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

Yes. Use modify.

As applied to your example:

router.get('/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);
        });
});

You can store your query in a variable, apply your conditional where clause and then execute it, like this :

router.get('/questions', function(req, res) {
  var query = knex('questions')
              .select('question', 'correct', 'incorrect')
              .limit(50);

  if(req.query.param == some_condition)
    query.where('somecolumn', req.query.param) // <-- only if param exists
  else
    query.where('somecolumn', req.query.param2) // <-- for instance

  query.then(function(results) {
    //query success
    res.send(results);
  })
  .then(null, function(err) {
    //query fail
    res.status(500).send(err);
  });
});