PM 2.5 & PM 10

That's a little vague, could you be more specific?

Maybe PM 2.5 is four times less than PM10.

Where do you get the readings from?

Since PM10 includes all particulates smaller than 10μm, it includes the 2.5PM count. "Can one reading be converted to another ?" - no.

(Then along comes AQI (Air Quality Index) and completely obscures the issue).

Of course there is!

Google "air quality".

Nope, see NIOSH standards on this.

PM 10 is bigger than PM 2.5, it was believed that when you inhale PM2.5 it wont be able to find its way out your lungs.--Respirable dust if I'm not mistaken,

PM10 size and above particulates could be filtered by your nose hair and mucous "nose gum"--sometimes slurry or non- newtonian fluid

NO! These relate to two completely separate clases of particulate emissions, specifically from cooling towers, and by default when filling out the EPA-42 yearly reports, this is done by multiplying factors against circulation rate, and cooling water TDS (total dissolved solids). This reports as PM10, particulates that fall under 10 micrometers in diameter.

However, it can be easily shown that when the TDS of the water becomes higher and higher, any mist released in the form of drift from the vapor plume of a cooling tower cannot in fact make a particle below 10 micrometers, because one cannot simply fit all that salt in to a sphere that size. In this viewpoint, PM10 will in fact decrease to practically nil once the TDS reaches that critical level, and it is not as salty as seawater when that point is reached.

EPA calculations do not allow for this effect, but make the PM10 directly proportional to TDS at all TDS.

To further answer the OP questions: NOPE, and NOPE, the numbers cannot be derived from each other, or bear any particular relation to the other, and they cannot be converted to the other.