If both columns are strings, you can concatenate them directly:
df["period"] = df["Year"] + df["quarter"]
If one (or both) of the columns are not string typed, you should convert it (them) first,
df["period"] = df["Year"].astype(str) + df["quarter"]
Beware of NaNs when doing this!
If you need to join multiple string columns, you can use agg:
df['period'] = df[['Year', 'quarter', ...]].agg('-'.join, axis=1)
Where "-" is the separator.