The Knights of the Black Fist in Phlan are the result of Zhentarim activity, but they are not presently connected.
Short Version
The Zhentarim conquered Phlan and the governor they installed was a worshiper of Bane and the leader of the Brothers of the Black Fist, which were eventually re-organized into the Knights of the Black Fist--which became the city's military. Later, the Zhent operations in the area were largely obliterated during the Shadowbane War, but the governor stayed in power in part by intentionally distancing himself from the Zhents.
The Knights of the Black Fist remained mostly-loyal to Bane, but they gradually shifted over to being more of "Phlan's Military" than "Bane's Military." Over time, their loyalty to the line of of the governor steadily decreased, and they have never sought to reconnect with the Zhentarim.
Detailed Version
The Zhentarim and Bane
The Zhentarim have long been associated with Bane, largely because their founder's (Manshoon) second-in-command was a Baneite Priest (Fzoul Chembryl) who tolerated the worship of no other gods within the organization. Manshoon, paranoid as he was, never really trusted Fzoul and disliked the influence that he had over the Black Network.
A few major events led to the Black Network drifting away from Bane's worship.
First was the Banedeath. In the year 1358, Bane was slain by Torm--most of his power went to Cyric, some went to Bane's son: Iyachtu Xvim [1]. Fzoul lost his position as High Priest, as the god he worshiped was no more, and the Zhents split their worship between the two new gods that held Bane's power. They did not get along. In 1361 a purge called the Banedeath began, with the followers of Cyric setting out to kill anyone who worshiped Iyachtu Xvim or still clung to Bane [2]...it was successful. Fzoul Chembryl was spared from the purge.
This rivalry ended up going on for a while, particularly coming to a head after Bane was resurrected in 1372 DR[1]. Control of the Zhents went back and forth a few times. In the end, Cyric's followers were victorious, in part because Fzoul Chembryl was slain in 1376DR by the Netherese when they destroyed Zhentil Keep (more on that below)[3]. They had solidified their power over the religious side of the Zhentarim.
Then the Spellplague happened and Cyric was sealed within the Supreme Throne for a thousand years for the murder of Mystra.[4] With the Baneites driven out (and mostly dead) and Cyric's clerics depowered, the religious leadership of the Zhentarim collapsed. And that brings us up to the present where the Zhents still have the same goals of wealth and domination, but are no longer directly tied to a particular deity.
Phlan
In the year 1375, Phlan was conquered by Zhentil Keep, which was the base of operations of the Zhentarim, and within 5 years their city council was replaced by a single ruler: Cvaal Daoran. Cvaal Daoran was a Hatemaster of the Zhentarim, a worshiper of Bane, and leader of the Order of the Black Fist--a religious order of knights dedicated in service to Bane [5]. (Note: This is after the first Banedeath)
When the Shadowbane war broke out in 1383, which destroyed most Zhentarim operations in the area, Daoran struck an alliance with the Fey in the nearby Quivering Forest, which ended up protecting Phlan from the Shadowvar until the war ended[6].
After the war ended, Cvaal Daoran declared his position to be hereditary and set about cutting ties with the Zhentarim...and the Zhents had largely been driven out of the region and what was left was crippled by the Shadowbane War, so they had no ability to protest this move.
(Note: the reason for his decision to distance himself from the Zhents is not explained, but it was most likely a move to solidify his power. As long as he was part of the Zhentarim, he answered to Zhent leadership and would get pulled into any conflict the Zhents got involved in--like the Shadowbane War.)
Roughly a century later, Cvaal's grandson (a cowardly and pampered noble) died in what was ostensibly a construction accident, and the Knight Commander of the Black Fist seized power--this has not been going well[6].
Final Notes
The Zhentarim have, somewhat recently, regained control of Zhentil Keep. As is their habit, they are likely looking to expand their influence, possibly hoping to reclaim a foothold within Phlan. It is possible that some Black Fist members are Zhentarim--but their membership is likely a secret.
References
1: Forgotten Realms Campaign Setting - 3rd Edition
2: Ruins of Zhentil Keep
3: Forgotten Realms Campaign Setting - 4th Edition
4: The Grand History of the Realms
5: Dungeon Magazine #170
6: Phlan during the Tyranny of Dragons
