The Chovos Halevovos's fourth שער, section, is on Bitachon. Since his sefer is "Duties of the heart", he clearly considers this a required "duty". Unlike some other areas, he does not have a sub-section where he gives the source for such a mitzvah.
There are many verses in Nach that do sound like they command Bitachon; some of them are at the end of Uva L'Tzion in the Shacharis davening.
בָּרוּךְ הַגֶּֽבֶר אֲשֶׁר יִבְטַח בה' וְהָיָה ה' מִבְטַחוֹ: בִּטְ֒חוּ בה' עֲדֵי עַד כִּי בְּיָהּ ה' צוּר עוֹלָמִים: וְיִבְטְחוּ בְךָ יוֹדְ֒עֵי שְׁמֶֽךָ כִּי לֺא עָזַֽבְתָּ דֹרְ֒שֶֽׁיךָ ה':
And more in Hallel:
יִ֭שְׂרָאֵל בְּטַ֣ח בה' עֶזְרָ֖ם וּמָגִנָּ֣ם הֽוּא׃
O Israel, trust in the LORD! He is their help and shield.
And many others.
But I don't think there is such a verse in the Chumash, and (as far as I know) it isn't listed as one of the 613 mitzvos. Is there a source in Chumash?