when you're on a ship full of MEN all day

"Partner", "partners", or "partnership" is said a total of 92 times in the show, not counting 11 times when it's part of a "previously on". Of those 92 instances, 11 are in Season One, 24 are in Season 2, 33 are in Season 3, and 24 are in Season 4. Some form of the word gets said at least once in all but eight episodes (1x01, 1x05, 3x01, 3x04, 3x09, 4x02, 4x03, and 4x05). The episode with most instances is 3x02 with a grand total of 7. In the video, the clips are arranged in chronological order according to when they appear in the show. I used the transcript site to double-check my work so I'm pretty confidant I didn't miss anything.

A lot of the time the relationships that get called partnerships also get referred to with other terms (friendship, alliance, husband/wife), either at different times than they're called partnerships or in conjunction with the term, but "partner" as a term interests me because it is non-specific in a way that allows for it to hold a lot of different meanings, sometimes all at once. Black Sails is clearly very specific about what terms are used for relationships, just as it is specific about what name a character is called by, and often there is significance placed on moments when a relationship is called a partnership. (Miranda and Flint in 2x06, Jack and Anne in 2x10, Flint and Sliver in 3x03, etc.)

SO I've made a full chart with details on each of the 92 instances which you can look at here. It has the episode each instance is from, who says it, and to what partnership they are referring. Generally this is pretty straightforward, though sometimes a character is directly referencing one partnership in a way that implicitly applies the label to another, i.e. when Silver says he's "not the first to have been a partner to [Flint] in this way", he is referring to himself as Flint's partner, but also implicitly stating that Gates and Miranda were each Flint's partner as well. Whenever this happens I've noted it in the chart. I've also noted when characters are speaking hypothetical partnerships that do not (yet) exist, or do not (yet) exist in the form they are currently discussed, and when partnerships are referred to in the past tense.

Highlights of some stuff I found interesting:

Silver and Flint are referred to as partners (or as having a partnership) a total of 7 times, and implicitly labelled as such once. This is more than any other partnership between individuals, and depending on how you count more than any partnership between groups, too (see below). The first time they're ever referred to as partners is by Silver in 3x03

Jack and Anne are just behind them with a total of 6, followed by Eleanor and Rogers with 5

The individual relationships between Jack and Max, and between Max and Anne are each referred to using some form of partner/partnership three times, and Jack, Anne, and Max a group of three are referred to using the term once by Max and once by Jack-- though when Jack uses it (while speaking to Marion Guthrie) he doesn't specify who he's speaking about could arguably also be including Featherstone, and possibly Idelle

The pirate and maroon alliance is referred to using partners/partnership 6 times (7 if you count one instance where Flint is used as synecdoche for the pirates as a group), and there are two instances of partner/partners being used when discussing another person or group joining the alliance (Jack in the first case, and Julius and his men in the second), for a total of 9 times that some form of the alliance is discussed using partners/partnership

Other than Flint, the only person who is ever referred to as Silver's partner is Max

Eleanor says partner (or some variation) more than any other character, for a total of 14 times. Runners up are Max (12) and Vane (9)

4 out of the 9 times Vane uses partner/partnership, he's talking about a partnership between Eleanor and someone, though that someone is not always him. Of the remaining 5 instances, two are in reference to his partnership with Teach and two involve Jack in some way (one with the two of them and Flint, and one in the context of Jack joining the pirate/slave alliance)

Max never uses the word partner to describe any relationships that she is not directly a part of, unless you count one instance of speaking broadly about her own experiences in a way that also includes Eleanor's (when she says the chair "demands you win partners" to get it)

Flint doesn't say any form of the word partner until 3x05, though before that point there are other people who call him their partner or apply the term to relationships he has

Thomas says "partner" a total of 3 times, two referring directly to Flint and one speaking generally but in a conversation about a partnership between him and Flint. Miranda says "partners" twice, both times about herself and Flint

"Partner" is used as a verb exactly once, by Madi