Mm, I don't think there is historical evidence of that, but there is in the other direction; diary entries from Murasaki Shikibu critical of Shonagon's writing style and calling her conceited.(IRL Shonagon was absolutely conceited. On a hilarious level. The Pillow Book is so entertaining).
I think it's safe to say they prolly didn't like each other, but I doubt they ever met since they served rival Empresses(The Emperor at the time got another Empress(Shoushi) and technically demoted Teishi in favor of her.)
Luckily everyone is friendly or more in this one and the Emperor is also a girl.