In the case the difficulty increases as much in future, will the possible hashes run out/be insecure?
As I understood, in the block hash, the amount of zero's is the nonce,


No, that is incorrect.

The nonce is a value in the block header, which is then hashed. By changing the nonce, a new block header is effectively created which has its own unique hash.


  Then the hash is recomputed for each value until a hash containing the required number of zero bits is found.


No. Contrary to popular belief, the correct hash is not based on the number of zeros in the block hash. That is just an abstraction to make it easier for people to understand.

What the correct hash actually uses is a target value. This target value is a 256 bit integer. In order for a block hash to be considered having a valid proof of work, it, when interpreted as a 256 bit integer must be less than or equal to the target value.

Since it is actually an integer comparison and not checking the number of 0 bits, there are far more possible target values.