The simplest way is to convert Date to POSIXlt format. Then perform the arithmetic operation as follows:
date_1m_fwd <- as.POSIXlt("2010-01-01")
date_1m_fwd$mon <- date_1m_fwd$mon +1
Moreover, incase you want to deal with Date columns in data.table, unfortunately, POSIXlt format is not supported.
Still you can perform the add month using basic R codes as follows:
library(data.table)
dt <- as.data.table(seq(as.Date("2010-01-01"), length.out=5, by="month"))
dt[,shifted_month:=tail(seq(V1[1], length.out=length(V1)+3, by="month"),length(V1))]
Hope it helps.