1. If you are asking someone about the possibility of doing something, the most natural way would be with the conjunctions 'if' or 'whether'.
I'd like to know if it is possible to ...
I'd like to know whether it is possible to ...
Both of the above are indirect questions.
2. If by 'I'd like to know..' you mean that you want to find something out for yourself or assure yourself that something is true, you can use 'that'.
'Why are you going to climb that mountain?'
'I'd like to know that it's possible.'
In this case, you can also omit the conjunction. You can say ''I'd like to know it's possible.'