Problems concerning tangents and quadrature have a long history predating the Newton/Leibniz formulation of calculus; indeed, they are amongst the oldest problems in mathematics. It seems reasonable to assume that the relationship between the two problems was “well” understood for some time prior to the calculus.
According to Carl Boyer, in his A History of Mathematics, Isaac Barrow’s Lectiones geometriae (1670) prominently feature both problems, and it is “likely” that he understood the relationship between the two. However, Boyer also points out that Barrow’s dislike of analytic methods were “scarcely conducive to analytic discoveries” and that it was at Newton’s insistence that an analytic treatment of this material was included in Barrow’s text - Newton was acting as Barrow’s editor. Boyes also notes that Barrow’s conservative adherence to geometric methods kept him from making effective use of this relationship.
Many figures prior to Barrow appear to have been in position of all the necessary theory to positively identify the relationship - Fermat, Huygens, Wallis, Gregory, for example.
When was the inverse relationship between tangents and quadrature/area first identified?