Can inflation show up in assets without necessarily being reflected in classic inflation indexes? I have been having an argument with a friend of mine who obtained an economics degree in college. I have been arguing that increased money supply from the fed has led to inflation in asset prices, despite 'inflation' not really showing up in things like CPI. He argues that since Fed economists use CPI, I am wrong to even consider any other metric since I am not smarter or more knowledgeable than the fed. Is he right? Does inflating asset prices not really count as inflation?
Yes inflating asset prices does not count as an inflation generally speaking although your friend's explanation is incorrect.
Fed is not an authority that decides how inflation is defined. Inflation is generally in academic literature simply defined as (Lebow &amp; Rudd, 2016)

Inflation measurement is the process whereby changes in the prices of individual goods and services are combined to yield a measure of general price change.

Assets are neither goods and services so they should not be considered at all if we talk about inflation generally.
This being said economists also sometimes use the inflation concept to measure change in price level of things other than goods and services. Price indexes that track asset prices and concept of asset price inflation does exist, but if an economic text, book, article etc talks about inflation they will mean inflation as defined above. When they are using other meanings of inflation economists will always add clarification like  asset price inflation etc. However, again if the discussion is just about inflation, without any other caveats it is about general goods and price inflation. Also this is not because Fed would decree so, this is just general economic terminology (Fed has power over methodology they use for their measure of inflation but they try to capture inflation as accurately as possible).