Like Genki, Quartet is a two-volume set of Japanese textbooks published by The Japan Times, but unlike Genki I and II, Quartet is designed for intermediate learners. Previously, the most common recommendation for students after finishing Genki was to jump into Tobira: Gateway to Advanced Japanese Learning Through Content and Multimedia. However, many have noted the steep difficulty increase this presented, as that book is intended for advanced learners. The Japan Times also publishes Tobira 1: Beginning Japanese and An Integrated Approach to Intermediate Japanese, though the latter was last revised in 2008. In this way, Quartet could be seen as almost a spiritual successor to An Integrated Approach, providing students a smoother transition from the beginner level Genki series into intermediate study than what was previously offered.
Overall, Quartet is a really nice stepping stone after Genki, with volume I being an easier transition into intermediate-level material than Tobira. There's very little hand-holding here like you might be used to if you’re coming from Genki, as the book is almost entirely in Japanese aside from the grammar explanations, similar to Tobira. But once you get acclimated to this, it becomes one of the primary strengths of the book, allowing a learner to practice their comprehension skills in a natural manner across all four domains.
While the bulk of the content is dedicated to the reading sections, it's not hard to see how well the comprehensive approach would work in a classroom or study group setting. While not all of the content in Quartet is created equal, the approach works, giving a motivated student in the right environment all they'd need to take their Japanese to the next level. But if your studying is entirely self-led, perhaps you should look elsewhere.