You can use the perpage package:
\documentclass{article}
\usepackage{amsthm,perpage,etoolbox}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\MakePerPage{definition}
\renewcommand{\thedefinition}{\theperpage.\arabic{definition}}
% We encourage TeX to break before a definition so the numbering will be correct
\BeforeBeginEnvironment{definition}{\goodbreak}
\usepackage[paperheight=60pt]{geometry} % just for the example
\begin{document}
\begin{definition}
First on first page
\end{definition}
\begin{definition}
Second on first page
\end{definition}
\newpage
\begin{definition}
First on second page
\end{definition}
\end{document}
This will assign numbers 1.1, 1.2 and 2.1
