Correct approach to 10pt quote size?

You asked,

Assuming that the thesis is written with 12pt is [footnotesize the] correct approach to obtain 10pt for quote?

Short answer: Yes. More precisely, it's one among several possible correct approaches to obtaining the desired formatting outcome.

Now on to a longer answer. The report document class defines the option 12pt as follows:

DeclareOption{12pt}{renewcommand@ptsize{2}}

whereby @ptsize is set to 2. Later on in report.cls, one finds these instructions:

ProcessOptions
input{[email protected]}

This (a) executes renewcommand@ptsize{2} and (b) inputs the file size12.clo as @ptsize is expanded to 2.

Next, in the file size12.clo, one finds the following instructions:

renewcommandnormalsize{%
   @setfontsizenormalsize@xiipt{14.5}%
   % (other stuff)
   }

and

DeclareRobustCommandfootnotesize{%
   @setfontsizefootnotesize@xpt@xiipt
   % (other stuff)
   }

Aside: @xpt and @xiipt expand to 10 and 12, respectively.

Long story short: If the report class is loaded with the option 12pt, the main font size -- normalsize -- is 12pt on 2.5 points of leading, while footnotesize works out to 10pt on 2 points of leading. ("Leading" is an old-fashioned (archaic??) expression, which refers to the thickness of the strips of lead that were inserted between lines of metal type.)

What to do with all this information?

• If your document has just one quote environment, it's perfectly alright to insert the instruction footnotesize right after begin{quote}.

  Aside: Note that footnotesize is a switch and doesn't take an argument. Thus, footnotesize{kant[1]} is no different from footnotesize kant[1]; the latter is preferable since it doesn't create any false impressions.

• If your document features several quote environments and if they must all be typeset at 10pt, I suggest you load the etoolbox package in the preamble and insert the instruction

  AtBeginEnvironment{quote}{footnotesize}
  

  to automate the process of switching to the required font size.

• If, for some reason, you take pride in making LaTeX execute only the absolutely minimal number of instructions needed to get a given job done, you might observe that footnotesize actually performs additional instructions -- such as changing the default amount of vertical whitespace that's inserted above and below displayed equations -- besides changing the font size to 10pt on 2pt of leading; it turns out that these additional instructions aren't needed here. Indeed, instead of running footnotesize, you could also run

  fontsize{10}{12}selectfont
  

  to get the desired font size.