The claim on the banners outside Curzon Street station buildings make this claim, but was it the first?
Curzon Street opened on April 9th 1838 and was the junction of two main lines, the London & Birmingham Railway and the Grand Junction Railway. Both the London & Birmingham Rly and GJR was built over time. The GJR reached Birmingham first with a station at Vauxhall opened in July 4th, 1837. This second line was extended to a station adjacent to the LBR station during January 1839.
Other passenger railways had opened earlier. The Liverpool and Manchester Railway was opened September 15th, 1830 with passenger termini at Crown Street, Liverpool and Liverpool Road Manchester. Their railway was a main line between cities and had various branches to Bolton, Wigan and Warrington.
London Euston the terminus of the London & Birmingham was opened in 1837, but it was not until the completion of Kilsby Tunnel (June 24th, 1838) that the line between Birmingham and London was complete.
It is difficult to justify the claim, if the Liverpool & Manchester is considered a main line, but from January 1839 Curzon Street was at the heart of the main line that linked London with the North West.