How to remove symbols from a column using awk

Using gsub:

awk '{gsub(/\"|\;/,"")}1' file
chr1    134901  139379  -   ENSG00000237683.5
chr1    860260  879955  +   ENSG00000187634.6
chr1    861264  866445  -   ENSG00000268179.1
chr1    879584  894689  -   ENSG00000188976.6
chr1    895967  901095  +   ENSG00000187961.9

If you want to operate only on the fifth field and preserve any quotes or semicolons in other fields:

awk '{gsub(/\"|\;/,"",$5)}1' file 

If your data is formatted exactly as shown (i.e. no other " or ; in other columns that need to be preserved), then you can simply use tr to remove these characters:

tr -d '";' < input.txt > output.txt

Using sed to remove all instances of '";': sed -i 's/[";]//g' file

To only remove from 5th column sed is probably not the best option.