The Midland Counties' Railway Company existed between 1839 and 1844, connecting Nottingham, Leicester and Derby with Rugby and thence, via the London and Birmingham Railway, to London [i]. Given that the Author was born in 1844; it seems that the line continued to be known as the "Midland Counties" after the company was incorporated into the Midland Railway in 1844

Reference:
[i] [https://en.wikipedia.org/wiki/Midland_Counties_Railway accessed (16Apr20)