Air quality is the composition of the atmosphere in our immediate surroundings
Those who speak of air quality are talking about the composition of the atmosphere in our immediate environment, the air that we and animals and plants breathe; and any positive or negative influence that this air composition has. Oxygen obviously has a positive influence, as most live forms on Earth depend on it.
Climate is the average 'weather' over a long period of time
When we talk about climate we are talking about longer periods of time, the average 'weather' over a long term period (30 years).
Locally, our planet can be divided into climate zones such as continental climate, maritime climate or mediterranean climate. The geography in the immediate vicinity has an important influence here (mountain ranges, rivers, coastal areas, latitude, etc.).
Globally, the following phenomena play an important role in climate:
The greenhouse effect in which greenhouse gases in the atmosphere trap heat from the sun is a good phenomenon in itself. It ensures that on average we have a pleasant, liveable temperature here on Earth. Climate change is about the enhanced greenhouse effect, about changes in the concentration of various gases in the atmosphere that cause it to warm up more rapidly.
The Earth's climate is affected by large volcanic eruptions, which are the main causes of aerosol emissions into the atmosphere. In addition, once every 100 to 200 years there is a major volcanic eruption that can affect temperature and climate over a long period of time.
Climate in the sense of climate change is therefore about the climate globally, not locally anymore.
Climate-related phenomena are determined by large-scale processes (in time and place) that have a major impact on the environment such as formation of deserts, drying up of lakes or melting of glaciers.
This also explains why scientists can make predictions on future climate change, along with its global and local impacts, whereas at the same time a 10-day weather forecast already appears to be very difficult.