[I should've posted this answer a long time ago, given that it was answered in the comments.] In order to extend the map $S^4\to \mathbf{H}P^\infty$ to a map from $J_2(S^4)$, the composite of $[\iota_4,\iota_4]:S^7\to S^4$ with the inclusion of $S^4$ into $\mathbf{H}P^\infty$ must be null. This is not true: the Whitehead product $[\iota_4, \iota_4]\in \pi_7(S^4)$ would have to be $2\nu$ to get the desired extension, but Equation 5.8 of Toda's composition methods book says that $[\iota_4, \iota_4] = \pm (2\nu - \Sigma \nu')$, where $\nu'\in \pi_6(S^3)$ is the Blakers-Massey element.

[I should've posted this answer a long time ago, given that it was answered in the comments.] In order to extend the map $S^4\to \mathbf{H}P^\infty$ to a map from $J_2(S^4)$, the composite of $[\iota_4,\iota_4]:S^7\to S^4$ with the inclusion of $S^4$ into $\mathbf{H}P^\infty$ must be null. This is not true: the Whitehead product $[\iota_4, \iota_4]\in \pi_7(S^4)$ would have to be $2\nu$ to get the desired extension, but Equation 5.8 of Toda's composition methods book says that $[\iota_4, \iota_4] = \pm (2\nu - \Sigma \nu')$, where $\nu'\in \pi_6(S^3)$ is the Blakers-Massey element.

As Gustavo Granja points out in the comments, there is a $p$-local analogue of this (with $p>2$): the map $S^4\to \mathbf{H}P^\infty$ extends to a map $J_{(p-1)/2}(S^4)\to \mathbf{H}P^\infty$, and the composite with the $(p+1)/2$-fold Whitehead product $[\iota_4, \cdots, \iota_4]: S^{2p+1}\to J_{(p-1)/2}(S^4)$ produces $\alpha_1\in \pi_{2p+1}(\mathbf{H}P^\infty)$. (See also this answer: Is $\mathbb{H}P^\infty_{(p)}$ an H-space?) This implies that there is no extension to a map $J_{(p+1)/2}(S^4)\to \mathbf{H}P^\infty$.