You can make a sol of nano alumina particle using acidic water with continuous stirring. Variety of acids is used for it such as Nitric acid, acetic acid, oxalic acid, depends on your material which suited best to you.
Coating can be done in various ways, if you have a porous monolith it's easy to use dipcoating or casting, if not then you could try spraying. It depends a lot on the shape and material of your piece.
As for the alumina stability, it's common to add Sodium Carbonate or Sodium Silicate as deflocculants, they will work as a barrier between particles, affecting viscosity. That does however interfere with your final chemical composition so keep that in mind if it's important for your application.
Been nano in size means a lot of headache, since the repulsion energy is proportional to the inverse of the particle size. Small diameters means they will regroup easily and faster. It's very hard to maintain a nanometer size suspension stable, don't give up but also don't except someone to come with a magic answer.
I stronglly suggest you read about Zeta Potential and Isoelectric point (IEP) as this can change your stability and viscosity.
Principles of Ceramics Processing, 2nd Edition, James S. Reed (1995) - read section IV (Processing addivitves).
Ceramic Processing, Mohamed N. Rahaman (2006) - read chapter 4 (SCIENCE OF COLLOIDAL PROCESSING ).
The link corresponds to an article about gamma alumina nano sized particles: Synthesis and characterization of nano-sized porous gamma-alumina by control precipitation method.