I have a 20 x 4000 dataframe in Python using pandas. Two of these columns are named Year and quarter. I\'d like to create a variable called p         
        
This solution uses an intermediate step compressing two columns of the DataFrame to a single column containing a list of the values. This works not only for strings but for all kind of column-dtypes
import pandas as pd
df = pd.DataFrame({'Year': ['2014', '2015'], 'quarter': ['q1', 'q2']})
df['list']=df[['Year','quarter']].values.tolist()
df['period']=df['list'].apply(''.join)
print(df)
Result:
   Year quarter        list  period
0  2014      q1  [2014, q1]  2014q1
1  2015      q2  [2015, q2]  2015q2
One can use assign method of DataFrame:
df= (pd.DataFrame({'Year': ['2014', '2015'], 'quarter': ['q1', 'q2']}).
  assign(period=lambda x: x.Year+x.quarter ))
Here is an implementation that I find very versatile:
In [1]: import pandas as pd 
In [2]: df = pd.DataFrame([[0, 'the', 'quick', 'brown'],
   ...:                    [1, 'fox', 'jumps', 'over'], 
   ...:                    [2, 'the', 'lazy', 'dog']],
   ...:                   columns=['c0', 'c1', 'c2', 'c3'])
In [3]: def str_join(df, sep, *cols):
   ...:     from functools import reduce
   ...:     return reduce(lambda x, y: x.astype(str).str.cat(y.astype(str), sep=sep), 
   ...:                   [df[col] for col in cols])
   ...: 
In [4]: df['cat'] = str_join(df, '-', 'c0', 'c1', 'c2', 'c3')
In [5]: df
Out[5]: 
   c0   c1     c2     c3                cat
0   0  the  quick  brown  0-the-quick-brown
1   1  fox  jumps   over   1-fox-jumps-over
2   2  the   lazy    dog     2-the-lazy-dog
Here is my summary of the above solutions to concatenate / combine two columns with int and str value into a new column, using a separator between the values of columns. Three solutions work for this purpose.
# be cautious about the separator, some symbols may cause "SyntaxError: EOL while scanning string literal".
# e.g. ";;" as separator would raise the SyntaxError
separator = "&&" 
# pd.Series.str.cat() method does not work to concatenate / combine two columns with int value and str value. This would raise "AttributeError: Can only use .cat accessor with a 'category' dtype"
df["period"] = df["Year"].map(str) + separator + df["quarter"]
df["period"] = df[['Year','quarter']].apply(lambda x : '{} && {}'.format(x[0],x[1]), axis=1)
df["period"] = df.apply(lambda x: f'{x["Year"]} && {x["quarter"]}', axis=1)
As many have mentioned previously, you must convert each column to string and then use the plus operator to combine two string columns. You can get a large performance improvement by using NumPy.
%timeit df['Year'].values.astype(str) + df.quarter
71.1 ms ± 3.76 ms per loop (mean ± std. dev. of 7 runs, 10 loops each)
%timeit df['Year'].astype(str) + df['quarter']
565 ms ± 22.3 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
Let us suppose your  dataframe is df with columns Year and Quarter.
import pandas as pd
df = pd.DataFrame({'Quarter':'q1 q2 q3 q4'.split(), 'Year':'2000'})
Suppose we want to see the dataframe;
df
>>>  Quarter    Year
   0    q1      2000
   1    q2      2000
   2    q3      2000
   3    q4      2000
Finally, concatenate the Year and the Quarter as follows.
df['Period'] = df['Year'] + ' ' + df['Quarter']
You can now print df  to see the resulting dataframe.
df
>>>  Quarter    Year    Period
    0   q1      2000    2000 q1
    1   q2      2000    2000 q2
    2   q3      2000    2000 q3
    3   q4      2000    2000 q4
If you do not want the space between the year and quarter, simply remove it by doing;
df['Period'] = df['Year'] + df['Quarter']