Can we use boosting algorithms like Adaboost and gradient boosting with only one classifier
Boosting typically only use one algorithm as it's base learner (almost exclusively decision trees). However, you could use a mixed set of algorithms as your base learners.

Something like this:

Boosting round 0: Add decision tree
Boosting round 1: Add neural network
Boosting round 2: Add KNN
Boosting round 3: Add decision tree
...


The reason you only see boosting using the same algorithm is probably just because it works better. I speculate that the diversity that comes from using several algorithms shine more when they are trained in parallel and combined. In boosting the base learners are trained in a sequence.