You cannot see all available channels, but you can leverage CSCC (Configuration System Chaincode) GetChannels API to get a list of channels client eligible to. All you need to do is invoke chaincode named CSCC.
The CSCC is the system chaincode, meaning it inherently built in into peer binary and loaded and "instantiate" during peer startup. Considering NodeJS the request will look as following:
const request = {
chaincodeId : "cscc",
txId: 213213123123, // Some random transaction id
fcn: "GetChannels",
args: ['']
}
it can also be done by the peer cli
peer channel list