The Five Factions
The Five Factions have been known to work together, so there isn't any implicit conflict there.
Each faction has its own motivations, goals, and philosophy. Some are more heroic than others, but all band together in times of trouble to thwart major threats.
The two additional factions added in Waterdeep: Dragon Heist are relatively specific to the Sword Coast (one operates in Waterdeep and regions north, the other in Waterdeep and regions below). They're cults of personality, lead by a single individual, rather than real factions in the same light as the big five.
The Emerald Enclave is...
[...] not opposed to civilization or progress, but they strive to keep it in balance with the wild.
As long as the Enclave member's actions don't actively harm the wilderness or swing the balance against it, they shouldn't have any specific problems. Individual Enclave contacts may frown upon it, but that's a personal objection, not an objection the organization as a whole would automatically support.
As long as the friends of a Zhentarim agent doesn't do anything to compromise other Zhentarim operations, they're not going to care. The Zhentarim...
[...] encourages individual ambition and rewards innovators who take matters into their own hands. Results are all that matter.
If some goofball tree hugger wants to help out a Zhentarim agent, so be it. Sounds like one of the few situations where somebody (the Zhentarim) got something for nothing.