My voice became a little high pitched due to my nervousness.
この場合、次のような言い方ができます。
ーMy voice sounded shrill because I was nervous.
「緊張して声が上ずった」
to sound shrill で「上ずる」を表現できます。
ーMy voice became a little high pitched due to my nervousness.
「緊張で声が上ずった」
one's voice become a little high pitched で「声がちょっとかん高くなる」=「声が上ずる」とも表現できます。
ご参考まで!