If it is possible to get someone to convert to our state religion, the name of if will appear underneath "Convert" in the trade table. Clicking on the word "Convert" itself will never do anything. For the option to be available though the following must be true:
1)You must be currently running a state religion.
2)The other Civ must currently be running a state religion that is not the same as yours.
3)Your religion must be presen in at least one of the other civ's cites.
Unless all three of these are true the option to convert them to your religion will not appear at all. Even if they are all true the other Civ may not necessarily be willing to convert, but in that case the name of your religion will be listed underneath "Convert" in red.
Similarly "Adopt" is merely the heading for the list of options, and doesn't do anything itself. To ask another civ to adopt your civics the following must be true:
1)You can only ask the AI to swap to a civic you are currently running.
2)The AI must not already be running the same civic
3)The AI must have the necessary technology to switch to that civic
4)The UN must not have passed the universal civic resolution for that column.
Unless all these are true, the option for them to switch civic will not appear under "Adopt".