You can perform quasi-perl-style string interpolation using the gsubfn package. 
library(gsubfn)
num <- 3
dataframe <- fn$sqlQuery(connection, 
         "SELECT dimension1 FROM table1 WHERE dimension1 = $num ")
Make use of paste or sprintf to construct the string:
sql <- paste("SELECT dimension1 FROM table1 WHERE dimension1 =", num)
sqlQuery(connection, sql)
or
sql <- sprintf("SELECT dimension1 FROM table1 WHERE dimension1 = %d", num)
sqlQuery(connection, sql)