While a number of modern martial arts groups claim that the Qing banned martial arts, it isn't true. There were restrictions on the general population owning weapons for the first few years of Qing rule (1644-1649) but the ban was ended in 1649, and confiscated weapons returned. Note that this was long before the Qing completed their conquest of China.

What the Qing did ban were some of the magico-religious-rebellious secret societies. Many of these societies included martial arts training as one of their activities, and where there was a signature martial arts style practiced by the society, training in that martial art could be used as evidence of membership of the banned society. Esherick (1987) discusses the link between the banned White Lotus society and Yi-he Boxing in the mid-Qing in an appendix, including discussion of cases where martial artists were investigated and accusations against them found to be baseless - training in martial arts was OK, but training in Yi-he would not have been OK (not strictly illegal, but certainly cause for suspicion of White Lotus membership). The prohibitions of these martial-arts practicing secret societies appears to be the origin of much of the myths claiming that the Qing banned martial arts.

Apart from the Qing not banning martial arts, they encouraged training in martial arts (for service to the state, not rebellion against the state). Traditionally, the most important martial arts had been weapons arts: archery, sword, and spear (Lorge, 2012). While the Qing army made heavy use of firearms in battle, the traditional weapons were still seen as important. While the Manchu emphasis on archery and horsemanship as an important part of Manchu identity wasn't as relevant to the non-Manchu population, the Qing encouraged candidates for the civil service examinations to learn martial arts, and at times required them to, including a practical test on mounted archery skills as part of the examination (the lower-status parallel military examination system had a large martial arts component, examining archery and other weapons skills). For the candidates to do well, they would need to train. This continued into the mid-19th century, when the need for modernisation of the army led to the traditional martial losing status (Huang and Hong, 2018).

There was also much dependence on local militias to keep order, to suppress banditry, to prevent and stop rebellion. These militias needed to be armed, and were expected to be able to use their weapons.

At the individual level, it was accepted by the Qing that people needed weapons for self-defence, and training in martial arts was acceptable.

From the late Ming onwards, we see a growing proliferation of martial arts, and especially of unarmed martial arts. This growth in martial arts continued, and expanded, during the Qing. There are two factors contributing to this. First, the publishing industry grew, and many books on martial arts were published - this provides evidence of the various martial arts. It is possible that there was no growth in martial arts at all, and that the apparent growth is simply more evidence of martial arts. However, more mention of unarmed martial arts outside such specialised martial arts literature means that it is probably a real growth. The other factor, which contributed to the development of unarmed martial arts was the growing dependence on firearms in battle. With modern firearms appearing in the mid-19th century, traditional martial arts, whether armed or unarmed, became increasingly irrelevant to warfare and other large-scale violence, and a significant part of martial arts training was expressly devoted to health (like most of modern Taiji/Tai Chi).

For more on Qing Dynasty martial arts and their roles in society, see the recent thesis by McNally (2019).

References:

Joseph Esherick, The Origins of the Boxer Uprising, University of California Press, 1987

Fuhua Huang and Fan Hong (eds), A History of Chinese Martial Arts, Routledge, 2018

Peter Lorge, Chinese Martial Arts: From Antiquity to the Twenty-First Century, Cambridge University Press, 2012

Ian McNally, Internal Cultivation or External Strength?: Claiming Martial Arts in the Qing Period, MA thesis, The Ohio State University, 2019